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CS 457Link-State and Distance
Vector Routing Fall 2012
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1
23
0111
value in arrivingpacket’s header
routing algorithm
local forwarding tableheader value output link
0100010101111001
3221
Routing vs. Forwarding
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u
yx
wv
z2
2
13
1
1
2
53
5
Graph: G = (N,E)
N = set of routers = { u, v, w, x, y, z }
E = set of links ={ (u,v), (u,x), (v,x), (v,w), (x,w), (x,y), (w,y), (w,z), (y,z) }
Network Graph Abstraction
Remark: Graph abstraction is useful in other network contexts
Example: P2P, where N is set of peers and E is set of TCP connections
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Network Graphs: Link Costs
u
yx
wv
z2
2
13
1
1
2
53
5 • c(x,x’) = cost of link (x,x’)
- e.g., c(w,z) = 5
• cost could always be 1, or inversely related to bandwidth,or inversely related to Congestion, or …
Cost of path (x1, x2, x3,…, xp) = c(x1,x2) + c(x2,x3) + … + c(xp-1,xp)
Routing question: What’s the least-cost path between u and z ?
A routing algorithm: an algorithm that finds least-cost path
or shortest path
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Routing Algorithm Classification
Global or decentralized?Global:• all routers have complete
topology, link cost info• “link state” algorithms
Decentralized: • router knows physically-
connected neighbors, link costs to neighbors
• iterative process of computation, exchange of info with neighbors
• “distance vector” algorithms
Static or dynamic?Static: • Manual configuration• When routes change very
slowly over time after human intervention
Dynamic: • When routes may change
quickly– periodic update– in response to link cost
changes– In response to link
failures
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Link-State Routing
Dijkstra’s algorithm• net topology, link costs
known to all nodes– accomplished via “link
state broadcast” – all nodes have same info
• computes least cost paths from one node (‘source”) to all other nodes– gives forwarding table for
that node• iterative: after k iterations,
know least cost path to k destinations
Notation:• c(x,y): link cost from node x
to y; = ∞ if not direct neighbors
• D(v): current value of cost of path from source to dest. v
• p(v): predecessor node along path from source to v
• N': set of nodes whose least cost path is definitively known
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Dijsktra’s Shortest Path Algorithm
1 Initialization: 2 N' = {u} 3 for all nodes v 4 if v adjacent to u 5 then D(v) = c(u,v) 6 else D(v) = ∞ 7 8 Loop 9 find w not in N' such that D(w) is a minimum 10 add w to N' 11 update D(v) for all v adjacent to w and not in N' : 12 D(v) = min( D(v), D(w) + c(w,v) ) 13 /* new cost to v is either old cost to v or known 14 shortest path cost to w plus cost from w to v */ 15 until all nodes in N'
Notation:• c(x,y): link cost from node x to y; =
∞ if not direct neighbors
• D(v): current value of cost of path from source to dest. v
• p(v): predecessor node along path from source to v
• N': set of nodes whose least cost path is definitively known
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Dijkstra’s Algorithm: Example
Step012345
N'u
D(v),p(v)2,u
D(w),p(w)5,u
D(x),p(x)1,u
D(y),p(y)∞
D(z),p(z)∞
u
yx
wv
z2
2
13
1
1
2
53
5
u,x 2,u 4,x 1,u 2,x ∞u,x,y 2,u 3,y 1,u 2,x 4,y
u,x,y,v 2,u 3,y 1,u 2,x 4,yu,x,y,v,w 2,u 3,y 1,u 2,x 4,y
u,x,y,v,w,z 2,u 3,y 1,u 2,x 4,y
V W X Y Z
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Dijkstra’s Algorithm Limitations
Algorithm complexity: n nodes• each iteration: need to check all nodes, w, not in N• n(n+1)/2 comparisons: O(n2)• more efficient implementations possible: O(mlogn)
Oscillations possible when link costs change:• e.g., link cost = amount of carried traffic
A
D
C
B1 1+e
e0
e
1 1
0 0
A
D
C
B2+e 0
001+e1
A
D
C
B0 2+e
1+e10 0
A
D
C
B2+e 0
e01+e1
initially… recompute
routing… recompute … recompute
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Distance-Vector Routing
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Distance Vector Algorithm (1)
The Bellman-Ford EquationDefine:
dx(y) := cost of least-cost path from x to y
Then:
dx(y) = min {c(x,v) + dv(y) }
where min is taken over all V, where V is a neighbor of x
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Bellman-Ford Example (2)
u
yx
wv
z2
2
13
1
1
2
53
5Note dv(z) = 5, dx(z) = 3, dw(z) = 3
du(z) = min { c(u,v) + dv(z), c(u,x) + dx(z), c(u,w) + dw(z) } = min {2 + 5, 1 + 3, 5 + 3} = 4
Node that achieves minimum is nexthop in shortest path ➜ forwarding table
B-F equation says:
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Distance Vector Algorithm (3)Definitions• Dx(y) = estimate of least cost from x to y• Distance vector: Dx = [Dx(y): y є N ]
Local information• Node x knows cost to each neighbor v: c(x,v)• Node x maintains Dx = [Dx(y): y є N ]• Node x also maintains distance vectors
received from its neighbors– Dv = [Dv(y): y є N ] for each neighbor v
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Distance Vector Algorithm (4)Basic idea: • Each node periodically sends its own
distance vector estimate to its neighbors• When a node x receives new DV estimate
from a neighbor, it updates its own DV using B-F equation:
Dx(y) ← minv{c(x,v) + Dv(y)} for each node y ∊ N
• Eventually, the estimate Dx(y) converges to the actual least cost dx(y)
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Distance Vector Algorithm (5)
Iterative, asynchronous: each local iteration caused by:
• local link cost change • DV update message from
neighbor
Distributed:• each node notifies
neighbors, but only when its DV changes– neighbors then notify their
neighbors if necessary
wait for (change in local link cost or msg from neighbor)
recompute estimates
if DV to any dest has
changed, notify neighbors
Each node:
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x y z
xyz
0 2 7
∞ ∞ ∞∞ ∞ ∞
from
cost to
from
from
x y z
xyz
0
from
cost to
x y z
xyz
∞ ∞
∞ ∞ ∞
cost to
x y z
xyz
∞ ∞ ∞7 1 0
cost to
∞2 0 1
∞ ∞ ∞
2 0 17 1 0
time
x z12
7
y
node x table
node y table
node z table
Dx(y) = min{c(x,y) + Dy(y), c(x,z) + Dz(y)} = min{2+0, 7+1} = 2 Dx(z) = min{c(x,y) +
Dy(z), c(x,z) + Dz(z)} = min{2+1, 7+0} = 3
32
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x y z
xyz
0 2 7
∞ ∞ ∞∞ ∞ ∞
from
cost to
from
from
x y z
xyz
0 2 3
from
cost tox y z
xyz
0 2 3
from
cost to
x y z
xyz
∞ ∞
∞ ∞ ∞
cost tox y z
xyz
0 2 7
from
cost to
x y z
xyz
0 2 3
from
cost to
x y z
xyz
0 2 3
from
cost tox y z
xyz
0 2 7
from
cost to
x y z
xyz
∞ ∞ ∞7 1 0
cost to
∞2 0 1
∞ ∞ ∞
2 0 17 1 0
2 0 17 1 0
2 0 13 1 0
2 0 13 1 0
2 0 1
3 1 0
2 0 1
3 1 0
time
x z12
7
y
node x table
node y table
node z table
Dx(y) = min {c(x,y)+Dy(y), c(x,z)+Dz(y)} = min {2+0, 7+1} = 2
Dx(z) = min {c(x,y)+Dy(z), c(x,z)+Dz(z)} = min {2+1, 7+0} = 3
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Example 2Initial Distances to your neighbors
A
B
E
C
D
Info atnode
A
B
C
D
A B C
0 7 ~
7 0 1~ 1 0
~ ~ 2
7
1
1
2
28
Distance to node
D
~
~2
0
E 1 8 ~ 2
1
8~
2
0
E
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E Receives D’s Routes
A
B
E
C
D
Info atnode
A
B
C
D
A B C
0 7 ~
7 0 1~ 1 0
~ ~ 2
7
1
1
2
28
Distance to node
D
~
~2
0
E 1 8 ~ 2
1
8~
2
0
E
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E Updates Cost to C
A
B
E
C
D
Info atnode
A
B
C
D
A B C
0 7 ~
7 0 1~ 1 0
~ ~ 2
7
1
1
2
28
Distance to node
D
~
~2
0
E 1 8 4 2
1
8~
2
0
E
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A Receives B’s Routes
A
B
E
C
D
Info atnode
A
B
C
D
A B C
0 7 ~
7 0 1~ 1 0
~ ~ 2
7
1
1
2
28
Distance to node
D
~
~2
0
E 1 8 4 2
1
8~
2
0
E
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A Updates Cost to C
A
B
E
C
D
Info atnode
A
B
C
D
A B C
0 7 8
7 0 1~ 1 0
~ ~ 2
7
1
1
2
28
Distance to node
D
~
~2
0
E 1 8 4 2
1
8~
2
0
E
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A Receives E’s Routes
A
B
E
C
D
Info atnode
A
B
C
D
A B C
0 7 8
7 0 1~ 1 0
~ ~ 2
7
1
1
2
28
Distance to node
D
~
~2
0
E 1 8 4 2
1
8~
2
0
E
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A Updates Cost to C and D
A
B
E
C
D
Info atnode
A
B
C
D
A B C
0 7 5
7 0 1~ 1 0
~ ~ 2
7
1
1
2
28
Distance to node
D
3
~2
0
E 1 8 4 2
1
8~
2
0
E
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Final Distances
A
B C
D
Info atnode
A
B
C
D
A B C
0 6 5
6 0 15 1 0
3 3 2
7
1
1
2
28
Distance to node
D
3
32
0
E 1 5 4 2
1
54
2
0
E
E
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Final Distances After Link Failure
A
B C
D
Info atnode
A
B
C
D
A B C
0 7 8
7 0 1
8 1 0
10 3 2
7
1
1
2
28
Distance to node
D
10
3
2
0
E 1 8 9 11
1
8
9
11
0
E
E
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View From a Node
A
B
E
C
D
dest
A
B
C
D
A B D
1 14 5
7 8 56 9 4
4 11 2
7
1
1
2
28
Next hop
E’s routing table
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The Bouncing Effect
A
25
1
1
B
C
B
C 21
dest costA
C 11
dest cost
A
B 12
dest cost
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C Sends Routes to B
A
25 1
B
C
B
C 21
dest costA
C 1~
dest cost
A
B 12
dest cost
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B Updates Distance to A
A
25 1
B
C
B
C 21
dest costA
C 13
dest cost
A
B 12
dest cost
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B Sends Routes to C
A
25 1
B
C
B
C 21
dest costA
C 13
dest cost
A
B 14
dest cost
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C Sends Routes to B
A
25 1
B
C
B
C 21
dest costA
C 15
dest cost
A
B 14
dest cost
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How Are These Loops Caused?
• Observation 1:– B’s metric increases
• Observation 2:– C picks B as next hop to A– But, the implicit path from C to A includes
itself!
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Solution 1: Holddowns
• If metric increases, delay propagating information– in our example, B delays advertising route– C eventually thinks B’s route is gone, picks
its own route– B then selects C as next hop
• Adversely affects convergence
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Other “Solutions”
• Split horizon– B does not advertise route to C
• Poisoned reverse– B advertises route to C with infinite
distance
Works for two node loops– does not work for loops with more nodes
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Example Where Split Horizon Fails
1
11
1
A B
C
D
• When link breaks, C marks D as unreachable and reports that to A and B.
• Suppose A learns it first. A now thinks best path to D is through B. A reports D unreachable to B and a route of cost=3 to C.
• C thinks D is reachable through A at cost 4 and reports that to B.
• B reports a cost 5 to A who reports new cost to C.
• etc...
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Avoiding the Bouncing Effect
Select loop-free paths• One way of doing this:
– each route advertisement carries entire path– if a router sees itself in path, it rejects the route
BGP does it this way
Space proportional to diameter
Cheng, Riley et al
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Distance Vector in Practice
• RIP and RIP2– uses split-horizon/poison reverse
• BGP/IDRP– propagates entire path– path also used for effecting policies
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LS vs. DV Algorithms
The two algorithms take complimentary approaches:
• Link State: Tell everyone what you know about your neighbors
• Distance vector: Tell your neighbors what you know about everyone
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Comparison of LS and DV Algorithms
Message complexity• LS: with n nodes, E links,
O(nE) msgs sent • DV: exchange between
neighbors only
Speed of Convergence• LS: O(n2) algorithm requires
O(nE) msgs– may have oscillations
• DV: convergence time varies– may have routing loops– count-to-infinity problem
Robustness: what happens if router malfunctions?
LS: – node can advertise
incorrect link cost– each node computes only
its own table
DV:– DV node can advertise
incorrect path cost– each node’s table used by
others • error propagates thru
network